Go for it OP.
Some of the comments here have been wildly off mark, and rude.
I know many parents who don't pay a bean for fees ( Grandparents,/ parents or trusts or using house price inflation from inherited property they got for nothing from their parents). Partial Bursary payments paid to middle classed parents on 70K income who are rich compared to you, but not rich compared to many others.
to a certain extent, the maze of applying for private schools , like applying to Oxford or applying for a top job in a media company, it is a HUGE advantage if you have friends and family 'in the know' what to prep your child for, when to register, when to apply, etc, etc, etc...so its true it is 'who you know' to a certain extent. But you can gain all this knowledge, but you must gain it in isolation if this isn't part of your clique and friends and family .
Have a list of schools , both state and private.
My advice would be to self tutor, get yourself in a stable position , all London primary schools are at least decent, a few teachers will at least TRY and help if you're aiming for 11+ entry schools, Read up on 11+ entry and try for the many schools that offer full bursary and try state grammar schools and semi selective schools that are very good also.
You don't have to pay for a tutor, you can tutor yourself, look on Ebay and buy used Bond books, etc.
Spend hours researching right here on MN, lots of entry advice and bursary advice also.
Most of all be realistic, it may happen, it may not, have a plan B, plan C, and Plan D, and make sure your DC are all in with the plan!
Best of Luck!!